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Q.

A ring and a solid sphere rotating about an axis passing through their centers have same radii of gyration. The axis of rotation is perpendicular to plane of ring. The ratio of radius of ring to that sphere is $\sqrt{\frac{2}{x}}$ . The value of x is ________

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answer is 5.

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Detailed Solution

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For ring $I = {MR}_{1}^{2}$
$K_{ring} = \sqrt{\frac{I}{M}} = R_{1}$
For solid sphere $I = \frac{2}{5} {MR}_{2}^{2}$
$K_{sphere} = \sqrt{\frac{I}{M}} = \sqrt{\frac{2}{5}} \cdot R_{2}$
In Given question K_ring = K_sphere
$\Rightarrow R_{1} = \sqrt{\frac{2}{5}} R_{2}, \Rightarrow \frac{R_{1}}{R_{2}} = \sqrt{\frac{2}{5}} = \sqrt{\frac{2}{x}} Ans : x = 5$

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A ring and a solid sphere rotating about an axis passing through their centers have same radii of gyration. The axis of rotation is perpendicular to plane of ring. The ratio of radius of ring to that sphere is 2x. The value of x is ________